\section{Analysis and Implementation Review}
\label{sec:review}

This section consolidates insights drawn from the recorded experiments and the current codebase structure. The goal is to bridge the gap between empirical observations and implementation decisions so that future debugging efforts can target the highest-leverage areas.

\subsection{Interpretation of Experimental Results}

\textbf{1D Helmholtz.} Table~\ref{tab:subdomains} shows that increasing the number of subdomains produces a dramatic error reduction between $N_e=1$ and $N_e=4$, after which the improvement saturates. The residual plateau at approximately $2.0$ suggests that the solver is converging to a biased solution rather than being under-resolved. Table~\ref{tab:training} confirms this behaviour: pushing the hidden-layer width from $M=50$ to $M=300$ yields more than an order-of-magnitude drop in error, but the asymptote remains far above the paper's $10^{-9}$ reference. Together these trends point to a systematic modelling discrepancy (likely in the operator assembly or scaling) rather than a lack of capacity or sampling.

\textbf{2D Helmholtz.} The timeout at the smallest tested configuration, despite using the same TensorFlow backend as the 1D case, indicates an algorithmic bottleneck. Profiling traces cited in the documentation show almost all time spent inside nested automatic-differentiation loops, validating that the implementation scales as $\mathcal{O}(M \times Q)$ per residual evaluation instead of the expected batched complexity.

\textbf{Cross-cutting signals.} The qualitative trends---error decreasing with richer discretisations and runtime scaling linearly with parameter counts---suggest that the overall architecture follows the paper's design. The quantitative mismatch (large bias in 1D, timeout in 2D) therefore likely stems from implementation subtleties rather than conceptual flaws.

\subsection{Implementation Assessment}

\textbf{Strengths.}
\begin{itemize}
  \item Clear modular decomposition (	exttt{Domain}, 	exttt{MultiSubdomainNetwork}, continuity conditions, and solver stack) closely mirrors the reference methodology.
  \item Documentation provides thorough guidance, with Markdown reports and helper scripts that make the debugging process reproducible.
  \item The linear solver follows a conventional least-squares pipeline, easing instrumentation and inspection of intermediate matrices.
\end{itemize}

\textbf{Weaknesses.}
\begin{itemize}
  \item Derivative computation in \texttt{helmholtz\_operator\_2d} is fully serial, negating the benefits of vectorised autodiff and leading to exponential slowdowns.
  \item Operator validation lacks automated tests; errors in sign conventions, scaling, or boundary enforcement can go undetected until late in the workflow.
  \item Conditioning diagnostics are not integrated into the solver, hampering the ability to distinguish between ill-posed systems and modelling errors.
  \item Experiment scripts couple data generation and plotting tightly, which complicates batch sweeps or notebooks for broader parameter exploration.
\end{itemize}

\subsection{Targeted Improvement Opportunities}

\begin{enumerate}
  \item \textbf{Operator sanity checks.} Introduce lightweight regression tests that evaluate the Helmholtz operators on analytical polynomials and compare the numerical residuals against symbolic derivatives. These checks would immediately expose sign or scaling mistakes contributing to the 1D bias.
  \item \textbf{Vectorised autodiff backend.} Refactor derivative assembly to use \texttt{tape.jacobian} or Hessian helpers so that all hidden units share the same tape evaluation. A prototype (as outlined in the bug summary) should reduce the cost from repeated forward passes to a single batched computation.
  \item \textbf{Conditioning monitors.} Record and log the condition number of the least-squares system and residual norms at solve time. Correlating these metrics with error trends can differentiate ill-conditioning from modelling issues and guide the choice of regularisation.
  \item \textbf{Experiment automation.} Separate parameter sweeps, raw result storage, and figure generation to streamline larger design-of-experiments campaigns. This also simplifies adopting alternative backends (e.g., JAX) for quick A/B comparisons.
  \item \textbf{Reproducible seeds and normalisation audit.} Trace the exact pseudorandom streams and input scaling employed by the original paper. Re-creating those heuristics---potentially via shared scripts from the authors---may resolve the residual offset without structural changes to the solver.
\end{enumerate}
